To solve this problem, we need to analyze each event and select the outcomes that meet the criteria described. Then, we calculate the probability of each event by dividing the number of favorable outcomes by the total number of outcomes (8, in this case).

Given Data:

The possible outcomes are:

• EOO, EOE, OOE, EEE, EEO, OEO, OOO, OEE

Each outcome represents a sequence of even (E) and odd (O) rolls in three trials.

Analysis of Each Event:

Event A: An even number on both the first and the last rolls

We need outcomes where the first and third positions are "E".

• Matching Outcomes: EOE, EEE, EEO
• Number of Favorable Outcomes: 3
• Probability: $\frac{3}{8}$

Event B: Exactly one odd number

We need outcomes with exactly one "O".

• Matching Outcomes: EOE, EEE, EEO
• Number of Favorable Outcomes: 3
• Probability: $\frac{3}{8}$

Event C: An even number on the first roll

We need outcomes where the first position is "E".

• Matching Outcomes: EOO, EOE, EEE, EEO, OEE
• Number of Favorable Outcomes: 5
• Probability: $\frac{5}{8}$

Summary Table:

Would you like further details or have questions?

Related Questions:

1. What is the probability of rolling an even number on all three rolls?
2. How many possible outcomes would there be if the cube were rolled four times?
3. What is the probability of having no odd numbers at all in the sequence?
4. If only odd rolls were counted as favorable, how would the probabilities change?
5. How does the probability of Event A change if we modify the conditions to “at least one even number”?

Tip:

To determine event probabilities, always check which outcomes meet the criteria and count them accurately before calculating.